The answer is simply no. The EU has extensive rules about import duties (a common tariff) and VAT, some rules about the “social security” system (health insurance, retirement) but almost nothing on income and other taxes. That's an area that has always been politically sensitive and extraordinarily complicated so that very little progress has been made towards harmonization.
EU countries do not even share the same tax year, let alone the same definition of income or the same type of taxes! So there are places where the tax year is distinct from the civil year (e.g. the UK), flat taxes, taxes on income that are really wealth taxes (e.g. “box 3” in the Dutch income tax), you name it.
Many countries will have generally sensible rules (it's seldom as simple as a 183 days threshold), even interpreted in isolation, and many others will have tax treaties to avoid any issues but none of that is guaranteed, even within the EU. So you really should not expect much and really need to look at the specifics of your situation.